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An Introduction to Supersingular Elliptic Curves and Supersingular Primes

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An Introduction to Supersingular Elliptic Curves and Supersingular Primes An Introduction to Supersingular Elliptic Curves and Supersingular Primes Anh Huynh Abstract In this article, we introduce supersingular elliptic curves over a finite field and relevant concepts, such as formal group of an elliptic curve, Frobenius maps, etc. The definition of a supersin- gular curve is given as any one of the five equivalences by Deuring. The exposition follows Silverman’s “Arithmetic of Elliptic Curves.” Then we define supersingular primes of an elliptic curve and explain Elkies’s result that there are infinitely many supersingular primes for every elliptic curve over Q. We further define supersingular primes as by Ogg and discuss the Mon- strous Moonshine conjecture. 1 Supersingular Curves. 1.1 Endomorphism ring of an elliptic curve. We have often seen elliptic curves over C, where it has many connections with modular forms and modular curves. It is also possible and useful to consider the elliptic curves over other fields, for instance, finite fields. One of the difference we find, then, is that the number of symmetries of the curves are drastically improved. To make that precise, let us first consider the following natural relation between elliptic curves. Definition 1. Let E1 , E2 be elliptic curves. An isogeny from E1 to E2 is a morphism φ: E E 1 → 2 such that φ( O) = O. Then we can look at the set of isogenies from an elliptic curve to itself, and this is what is meant by the symmetries of the curves. Definition 2. Let E be an elliptic curve. Let End( E) = Hom( E, E) be the ring of isogenies with pointwise addition ( φ + ψ)( P) = φ( P) + ψ( P) and multiplication given by composition 1 ( φψ)( P) = φ( ψ( P)) . Then End( E) is called the endomorphism ring of E. 1.2 Classification of the endomorphism rings. The first question we can ask is what kind of ring can this endomorphism ring be. The following objects are needed in the formulation of the answer. Definition 3. (Order of an algebra) Let be an algebra finitely generated over Q. An K order of is a subring of which is finitely generated as Z-module and which satisfies R K K Q = . R ⊗ K Example 4. Let be the quadratic imaginary field Q √ D for some positive D, its K − O ring of integers. Then for each integer f > 0, the ring Z + f is an order of . O K Definition 5. A (definite) quaternion algebra over Q is an algebra of the form = Q + Qα + Q β + Qαβ K with the multiplication rules α2 ,β2 Q, α2 < 0,β2 < 0, βα = αβ. ∈ − Then we have the following theorem. Theorem 6. (Classification of endomorphism ring) The endomorphism ring of an elliptic curve is either Z (rank 1), an order in a quadratic imaginary field Q √ D (rank 2), or an order in a quaternion algebra (rank 4). − (See Silvermans’s “Arithmetic of the Elliptic Curves” , Chapter III, Section 9 for a proof.) If char( K) = 0, then End( E) Q cannot be a quaternion algebra (see remark to Theorem 12). For example, in class all⊗ we have seen so far are elliptic curves over C, so they only have endomorphism ring as Z or an order in a quadratic imaginary field. Example 7. The curve C/Z[ i] has endomorphism ring Z[ i] . The curve C/Z[ √ 2 ] has − endomorphism ring Z. If K is a finite field, however, then End( E) is always larger than Z. In fact, the particular class of elliptic curves that have endomorphism ring as an order in a quarternion algebra (the maximum possible) are called supersingular elliptic curves, the main object of this exposition. 2 1.3 Definition of a supersingular curve. As alluded to before, a supersingular elliptic curve is one that has the maximum number of symmetries: End( E) is an order in the quaternion algebra. In his 1941 paper, Deuring proved the equivalence of this and four other conditions, therefore each of which can then be considered as a definition for a supersingular curve. The main goal of this section is to state the theorem. We will need the following objects. Definition 8. Let R be a ring. A (one-parameter commutative) formal group defined over R is a power series F( X,Y) R[[ X,Y]] satisfying: F ∈ a ) F( X,Y) = X + Y + (terms of degree 2 ). b ) F( X,F( Y,Z)) = F( F( X,Y) ,Z) . ≥ c ) F( X,Y) = F( Y,X) d ) There is a unique power series i( T) R[[ T]] such that F( T , i( T)) = 0. e ) F( X , 0) = X,F(0,Y) = Y. ∈ In particular, for an elliptic curve E given by a Weierstrass equation with coefficients in R, 2 3 2 E: y + a1 xy + a3 y = x + a2 x + a4x + a6, the formal group associated to E, denoted by Eˆ , is given by the power series 2 2 3 2 2 3 F( z , z ) = z + z a z z a ( z z + z z ) (2a z z ( a a 3a ) z z + 2a z z ) + 1 2 1 2 − 1 1 2 − 2 1 2 1 2 − 3 1 2 − 1 2 − 3 1 2 3 1 2 ∈ Z[ a1 , , a6 ][[ z1 , z2 ]] . Definition 9. Let R be a ring of characteristic p > 0. Let , /R be formal groups and f: F G a homomorphism defined over R. The height of f, denoted by ht( f) , is the largest F → G h integer h such that f( T) = g Tp for some power series g( T) R[[ T]] . If f = 0 then ∈ ht( f) = . The height of , denoted ht( ) , is the height of the multiplication map [ p]: . ∞ F F F→F Example 10. If m 1 is prime to p, then ht([ m]) = 0, because [ m] T = mT + . ≥ Definition 11. (Frobenius map) Let K be a field of positive charateristic p, and let q = pr. Let f ( q) be the polynomial obtained from f by raising each coefficient of f to the q-th power. Then for each curve C/K we can define a new curve C( q) /K by describing its homogeneous ideal as I C( q) = ideal generated by f ( q) : f I( C) . ∈ The q-th power Frobenius morphism is the natural map from C to C( q) given by φ: C C( q) → q q φ([ x0, , xn]) = [ x0 , , xn] . 3 Definition 12. Let φ: C1 C2 be a map of curves defined over K. If φ is constant, we define the degree of φ to be →0; otherwise we say that φ is finite and define its degree by deg φ = [ K( C1 ): φ∗ K( C2 )] . We say that φ is separable, inseparable, purely inseparable if the extension K( C1 )/ φ∗ K( C2 ) has the corresponding property. In particular, the Frobenius map is inseparable. Now we can state Deuring’s theorem. Theorem 13. (Deuring) Let K be a perfect field of characteristic p and E/K an elliptic curve. For each integer r 1 , let ≥ ( pr ) ( pr ) φr: E E and φˆr : E E → → be the pr-th power Frobenius map and its dual. a ) The following are equivalent. i. E[ pr] = 0 for one (all) r 1 . ≥ ii. φˆr is (purely) inseparable for one (all) r 1 . ≥ iii. The map [ p]: E E is purely inseparable and j( E) F p2 . iv. End( E) is an order→ in a quaternion algebra. ∈ v. The formal group Eˆ /K associated to E has height 2. b ) If the equivalent conditions in (a) do not hold, then E[ pr] = Z/ prZ for all r 1 , ≥ and the formal group Eˆ /K has height 1 . Further, if j( E) F¯ p, then End( E) is an order in a quadratic imaginary field. ∈ (See Silverman’s “Arithmetic of Elliptic Curves” Chapter V, Section 3 for a proof.) Remark 14. 1. There are further definitions, for instance in terms of sheaf cohomology and residues of differentials. 2. Note that a supersingular curve is not singular. By definition it is an elliptic curve, hence smooth. 3. Item ( ii) explains what we claimed above that End( E) cannot be an order in a quaternion algebra if the curve over a C: for a field to possess a non-trivial purely inseparable extension, it cannot be perfect. An example of a supersingular elliptic curve is the curve X1 ( 11 ) , reduced modulo the prime 19. (See Section 2.1). 2 Supersingular Primes. Closely related to the concept of a supersingular elliptic curve is that of a supersingular prime. There are two different definitions, each refer to a different thing. 4 2.1 Supersingular Primes for an elliptic curve over Q. Definition 15. (Supersingular primes for an elliptic curve) If E is an elliptic curve defined over the rational numbers, then a prime p is supersingular for E if the reduction of E modulo p is a supersingular elliptic curve over the residue field F p. In his 1987 paper, Elkies shows that there are infinitely many such primes for each elliptic curve over Q. The idea is as follow. The reduction Ep of an elliptic curve E at a good prime p is supersingular if and only if it has complex multiplication by some order OD = Z 1 D + √ D ( D 0 or 3 mod 4) , such that p is ramified or inert in Q √ D . 2 − ≡ − Note that for each D, however, there are only finitely many isomorphism classes of elliptic curves over Q¯ with complex multiplication by OD . Furthermore, the j-invariants of these curves are conjugate algebraic integers.
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